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number-manifold of the kind described above, and this con-
ception of space grew out of Riemann* s 'manifolds'.

Passing on to measurement, Riemann states that *Measure-
ment consists in a superposition of the magnitudes to be com-
pared. If this is lacking, magnitudes can be compared only
when one is part of another, and then only the more or less, but
not the how much, can be decided.' It may be said in passing
that a consistent and useful theory of measurement is at present
an urgent desideratum in theoretical physics, particularly in all
questions where quanta and relativity are of importance.

Descending once more from philosophical generalities to less
mystical mathematics, Riemann proceeded to lay down a defi-
nition of distance, extracted from his concept of measurement,
which has proved to be extremely fruitful in both physics and
mathematics. The Pythagorean proposition

that a = fc2 + c2 or a = Vd2 + c2, where a is the length of the
longest side of a right-angled triangle and b,c are the lengths of
the other two sides, is the fundamental formula for the measure-
ment of distances in a plane. How shall this be extended to a
curced surface^ To straight lines on the plane correspond geode-
sies (see chapter 14) on the surface; but on a sphere, for
example, the Pythagorean proposition is not true for a right-
angled triangle formed by geodesies. Riemann generalized the
Pythagorean formula to any manifold as follows:
Let (2^2, ... , osn)9 (a?! + x^3 ai2 + xz', ... , xn -f *') be the
co-ordinates of two 'points' in the manifold which are 'infinite-
simally near' one another. For our present purpose the meaning
of 'infinitesimally near' is that powers higher than the second of
  - 9 &n> which measure the 'separation* of the two